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GRADE SUPPLEMENT Set A9 Number & Operations: Multiplying Fractions Includes Activity 1: Geoboard Perimeters Activity 2: Fraction Multiplication Story Problems Activity 3: Using the Area Model for Multiplying Fractions Activity 4: Generalizations About Multiplying Fractions Activity 5: Target 1: Fractions Activity 6: Multiplying Domino Fractions Activity 7: Area Word Problems with Mixed Numbers Independent Worksheet 1: Picturing Fraction Multiplication Independent Worksheet 2: More Fraction Multiplication Independent Worksheet 3: Fraction Stories Independent Worksheet 4: Using Strategies to Multiply Fractions with Mixed Numbers H Independent Worksheet 5: Domino Multiplication H H H H H H H H H H H A9.1 A9.9 A9.17 A9.23 A9.31 A9.37 A9.45 A9.49 A9.51 A9.53 A9.55 A9.57 Skills & Concepts H Add fractions with unlike denominators H Find the perimeter of regions with an area smaller than one H Estimate the results of operations performed on fractions and use the estimate to determine the reasonableness of the inal answer H Find the product of two unit fractions with small denominators using an area model H Multiply fractions using the standard algorithm H Explain the relationship of the product relative to the factors when multiplying fractions H Add mixed numbers with unlike denominators H Subtract mixed numbers with unlike denominators H Multiply a whole number by a fraction H Interpret multiplication as scaling (resizing) H Solve word problems involving multiplying fractions and mixed numbers using visual fraction models and equations P201304 Bridges in Mathematics Grade Supplement Set A9 Number & Operations: Multiplying Fractions The Math Learning Center, PO Box 12929, Salem, Oregon 97309 Tel 800 575–8130 © 2013 by The Math Learning Center All rights reserved Prepared for publication on Macintosh Desktop Publishing system Printed in the United States of America P201304 The Math Learning Center grants permission to classroom teachers to reproduce blackline masters in appropriate quantities for their classroom use Bridges in Mathematics is a standards-based K–5 curriculum that provides a unique blend of concept development and skills practice in the context of problem solving It incorporates the Number Corner, a collection of daily skill-building activities for students The Math Learning Center is a nonproit organization serving the education community Our mission is to inspire and enable individuals to discover and develop their mathematical conidence and ability We offer innovative and standards-based professional development, curriculum, materials, and resources to support learning and teaching To ind out more, visit us at www.mathlearningcenter.org Set A9 H Activity ACTIVITY Geoboard Perimeters Overview You’ll need In preparation for using the area model to multiply one fraction by another, students investigate the perimeter of the largest square that can be formed on the geoboard, as well as the perimeters of smaller regions on the geoboard H Rectangle Review (page A9.6, run for display) Skills & Concepts H geoboard and geobands (class set plus for display) H add fractions with unlike denominators H ind the perimeter of regions with an area smaller than H Geoboard Perimeters (page A9.7, run for display) H More Geoboard Perimeters (page A9.8, run a doublesided class set, plus a few extra) H pens H 2–3 blank transparencies H a piece of paper to mask portions of the display H 53 ⁄4˝ × ⁄4˝ strips of red construction paper (10–12 per student) H tile and red linear units available as needed H pencils and scissors Note When you represent the symbolic form for a fraction, please use a horizontal bar Instructions for Geoboard Perimeters Open the activity by explaining to the class that you are going to start a series of lessons on multiplying fractions To get started, you are going to review the area model for multiplication Then display the Rectangle Review master Review the information together, and ask students to pair-share responses to the questions: •฀ What is the area of the rectangle on the display? •฀ What information you need in order to determine the area of the rectangle? Have a few volunteers share their thinking with the class As the discussion proceeds, guide students to review the connection between perimeter, area, and multiplication Students We think it’s about 28 square inches We said it could be maybe be about 150 square centimeters We can’t tell, because we don’t know how long the sides are We don’t even know if they’re in inches or centimeters Teacher Why you need to know the side lengths to find the area of the rectangle? Students Because you get area by multiplying length times width You need to know how many squares will fit into the rectangle Like, if we know that squares fit © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.1 Set A9 Number & Operations: Multiplying Fractions Activity Geoboard Perimeters (cont.) across the top, and squares fit along the side, we would know the area is times 7, and that’s 28 But it depends on the size of the squares If they’re little, like square centimeters, the area could be more than 100 After some discussion, have a volunteer come up to the display and measure the side lengths of the rectangle in inches Then work with input from the class to label the rectangle and summarize students’ comments on the display Set A9 Number & Operations: Mu tip ying Fractions Blackline Run one copy for disp ay Rectangle Review What is the area of this rectangle? 4” 6” x = 24 square inches What information ytou need beofre you can answer the question? • units (inches, centimeters, or ?) • side lengths • then multiply the side lengths to get the area How are perimeter, are and multiplication related? • • • • You have to multiply to find area You have to know the lengths of the sides to find the area If you know the side lengths, you can find the perimeter If you know the area of a rectangle and the length of one side, you can find the length of the other side by dividing • A rectangle gives you a way to make a picture of multiplication Next, display the top portion of the Geoboard Perimeters master as helpers give students each a geoboard and some geobands Read the information on the display together and ask students to replicate the square on their own geoboard If the area of that square is unit, what is the length of each side, and what is the perimeter of the square? Give students a minute to pair-share ideas, and then call for and record their answers Set A9 Number & Operations: Mu tip ying Fractions Blackline Run one copy for d sp ay Geoboard Perimeters Jason says that the perimeter of this square is linear units Do you agree with him? Why or why not? Area = Square Unit Teacher Now that you’ve had a minute to think about the question, let’s record your answers here on the whiteboard What did you decide? A9.2 ã Bridges in Mathematics Grade Supplement â The Math Learning Center Activity Geoboard Perimeters (cont.) Students We don’t agree with Jason We think the perimeter of that square is 16 That’s what we got too We agree with Jason We think the perimeter is After you have recorded students’ answers, invite individuals or student pairs to the display to demonstrate their thinking Set a blank acetate on top of your master and then re-position it as needed, so that several different students can mark on it to show how they determined the perimeter of the square in question Teacher Any different ideas? No? Who’d like to convince us of their reasoning? You can mark on the display to show what you did to get your answer Jon We said it was 16 instead of We started in the corner of the board and just counted the pegs all the way around It came out to 16 10 11 12 16 15 14 13 Ariel We did kind of the same thing as Jon and Omid, but we looked at the spaces instead of the pegs It looked like each side of the square was 4, and we know that × is 16, so we said the perimeter of the square is 16 Gabe We think the perimeter is We said if the area of the whole square is 1, then each side must be So that means the perimeter of the square is 4, like this: 1, 2, 3, 4 Jasmine We agree with Gabe and Raven See, if each of the little squares was worth 1, then the perimeter would be 16, but the big square is worth 1, so each of the sides must be When students have had adequate time to discuss and debate the perimeter of the largest square, build the square on your own geoboard at the display and show one of the strips of red construction paper you have cut, first holding it up for all to see, and then setting it into the space between the edge and the pegs of the board Then invite students’ comments Teacher I cut some strips for us to use in considering the perimeter of this square What you think? © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.3 Set A9 Number & Operations: Multiplying Fractions Activity Geoboard Perimeters (cont.) Students Those are like the little red pieces we use with the tiles sometimes It’s like a giant red piece But those little red pieces are worth 1, so this one must be worth Teacher How are you thinking about that? Kamil Well, it goes along spaces on the geoboard, so it must be worth Hanako But that’s what we were trying to tell you before That square has an area of It’s like giant tile, and that strip is like giant red piece Confirm the fact that the red strips you have cut are each worth linear unit That being the case, what is the perimeter of the largest square on the geoboard? (4 linear units) Now display the middle portion of the master, which establishes that the perimeter of the largest square is linear units and asks students to determine the perimeter of several different regions on the geoboard If the biggest square on the geoboard has a perimeter of linear units, what is the perimeter of each lettered region? B A C D E Perimeter = Linear Units Work with the class to determine the perimeter of Region B Ask students to remove the large square from their board and build just Region B, as you place a handful of red construction paper strips at each table or cluster of desks Give students a few minutes to experiment with their strips as they consider the perimeter of this region Let them know that it is fine to fold and cut the strips if that helps them think about the length of each side of Region B Then invite or individuals or pairs to the display to share their thinking Ask them to work with a board and strips so their classmates can see what they are talking about as they explain Theo We were pretty stuck at first, but we kept looking at the strips and the rectangle on our board Then we realized that if you fold one of the strips in half, it fits along the top of the rectangle Then we knew that the long sides were each worth 1/2 Ichiro We found out that the small sides are each worth 1/4 of a linear unit If you fold one of those strips in half and then in half again, you get fourths If you cut them up, they fit right along the short sides of the rectangle, like this A9.4 • Bridges in Mathematics Grade Supplement © The Math Learning Center Activity Geoboard Perimeters (cont.) Kendra We did the same thing, and then we added them up because that’s what you when you’re figuring out the perimeter We got that it was 1/2, and that seems kind of weird Can you have a perimeter with a fraction in it? 10 As students share their thinking, use the lower portion of the master to label and record the dimensions of Region B When it has been established that the long sides are each 1/2 of a linear unit, and the short sides are 1/4 of a linear unit, work with student input to add the fractions to determine the total perimeter They will find, in fact, that the perimeters of some, though not all, of the regions are mixed numbers B + =1 1 + 4=2 1 + = linear units 11 Now give students each a copy of More Geoboard Perimeters (shown below with the answers and sample responses filled in for your reference) Ask students to sketch Region B, label the length of each side, and record one or more number sentences to show the computations necessary to find the total Then have them find the area of each of the other regions shown on the master: A, C, D, and E DATE More Geoboard Perimeters B 4 A C 4 2 1 1 linear units P = _ 2 linear units P = _ D E 4 P = _ linear units 4 4 P = _ linear units 4 4 P = _ linear units Set A9 Number & Operations: Multiplying Fractions Blackline Run a c ass set NAME 2 P = _ linear units Extension Students who determine and record the perimeters of all regions quickly and easily can be asked to build at least two figures (other than any of the regions they’ve already investigated) that have a perimeter of linear units, two that have a perimeter of 21/2 linear units, two with P = linear units, and two with P = 1/2 linear units Each discovery should be recorded the same way the first regions have been, using the last box on the record sheet, as well as the back of the sheet and a second sheet if necessary © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.5 Set A9 Number & Operations: Multiplying Fractions Blackline Run copy for display NAME DATE Rectangle Review What is the area of this rectangle? What information you need before you can answer the question? How are perimeter, area and multiplication related? A9.6 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline Run copy for display NAME DATE Geoboard Perimeters Jason says that the perimeter of this square is linear units Do you agree with him? Why or why not? Area = Square Unit If the biggest square on the geoboard has a perimeter of linear units, what is the perimeter of each lettered region? B A C D E Perimeter = Linear Units B © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.7 DATE More Geoboard Perimeters B A © The Math Learning Center P = _ linear units P = _ linear units D E P = _ linear units P = _ linear units C P = _ linear units P = _ linear units Set A9 Number & Operations: Multiplying Fractions Blackline Run a class set A9.8 • Bridges in Mathematics Grade Supplement NAME Set A9 Number & Operations: Multiplying Fractions Blackline Optional, run as needed NAME DATE Journal Page Grid A9.44 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 H Activity ACTIVITY Area Word Problems with Mixed Numbers Overview You’ll Need During this session, students will solve several word problems designed to help them develop eficient strategies for multiplying fractions by a whole number using an area model Their work will include situations involving improper fractions and mixed numbers The story problem context, along with the use of visual models, will help students make sense of the magnitude of the product H Area Word Problems with Mixed Numbers (page A9.47 run copy for display.) H Student Journals or Journal Page Grid (page A9.48 optional, run as needed.) Note When you represent the symbolic form for a fraction, please use a horizontal bar Skills & Concepts H Add fractions with unlike denominators, including mixed numbers (5.NF.1) H Multiply a whole number by a fraction (5.NF.4.) H Interpret multiplication as scaling (resizing) (5.NF.5) H Solve word problems involving multiplying fractions and mixed numbers using visual fraction models and equations (5.NF.6) Instructions for Area Word Problems with Mixed Numbers Open the session by telling students that today their work multiplying fractions will extend to mixed numbers and improper fractions Display the Area Word Problems with Mixed Numbers page so only the first word problem is showing and read it together Set A9 Number & Operations: Multiplying Fractions B ackline Run copy for d splay NAME DATE Area Word Problems With Mixed Numbers Here are five problems for you to solve For each one, • write the problem number in your journal • record an estimate (nearest whole number) and explain how you got it • make a labeled sketch to show your thinking • write a multiplication equation to match, including the answer A rectangle is meters long and meter wide What is the area? Invite students to make a sketch to show how they might find the area of this rectangle After a few minutes, ask them to share their model with a partner Finally, choose a student who correctly sketched and labeled an area model to share © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.45 Set A9 Number & Operations: Multiplying Fractions Activity Area Word Problems with Mixed Numbers (cont.) Set A9 Number & Operations: Multiplying Fract ons Blackl ne Optional Run cop es as needed Journal Page Grid ☎ ✂ ✄ ✄ x4= =2 Josie I shaded 1/2 of four boxes in the grid Next I added 1/2 four times for a sum of The area of the rectangle is 2m2 1/2 × = 4/2 =2 Then, show the rest of the Area Word Problems page and review the directions Depending on the strengths and needs of your class, you may want to have students one or two more problems with you, and then work in pairs Allow students who are ready to work independently to so while you work with those who need additional support As students work, watch for strategies to share like scaling up from a unit fraction or using doubling, repeated addition, or money or decimals Are students able to estimate and then compare the size of the products without having to the actual computation? Can they explain what happens to the product when the fraction is greater than or what happens when the fraction is less than 1? (the product increases or decreases respectively) The problems on the Area Word Problems with Mixed Numbers page lend themselves to the following strategies: •฀ doubling, •฀ repeated addition •฀ money/decimals •฀ scaling up from unit fraction With about 15 minutes left, call the class back together to discuss several of the problems Choose one of the problems most students finished and invite several students to explain their thinking If you have time, repeat with another problem, trying to showcase several strategies for each As you examine strategies for multiplying fractions, consider comparing and contrasting the procedures for adding, subtracting, and multiplying fractions Why does the algorithm work when we multiply the numerators and denominators across, when that algorithm doesn’t work with addition or subtraction of fractions A9.46 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline Run copy for display NAME DATE Area Word Problems With Mixed Numbers Here are five problems for you to solve For each one, •฀ write฀the฀problem number in your journal •฀ record฀an฀estimate (nearest whole number) and explain how you got it •฀ make฀a฀labeled sketch to show your thinking •฀ write฀a฀multiplication equation to match, including the answer A rectangle is meters long and meter wide What is the area? A painting in the county fair measures meters by 4 meters What is the area of the painting? A teacher measured his classroom door and found that it was 1 meters wide and meters tall What’s the area of the door? The rectangular top of a table is three times as long as it is wide Its width is meters Find the area of the table-top A small city park consists of a rectangular lawn that is 30 long and 20 meters wide What is the area of the lawn? Kale built a backyard pen for his new puppy The length of the pen is meters and the width is meters What is the area of the pen? © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.47 Set A9 Number & Operations: Multiplying Fractions Blackline Optional, run as needed NAME DATE Journal Page Grid A9.48 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline For use anytime after Supplement Set A9, Activity NAME DATE Set A9 H Independent Worksheet INDEPENDENT WORKSHEET Picturing Fraction Multiplication Each of the pictures below shows the results of multiplying one fraction by another Label each of the shaded regions with its dimensions and area Then write a multiplication equation to match ex a 2x = = 12 b c Pedro’s dog, Oso, got into the kitchen last night Oso saw three-fourths of a meat loaf still in the pan He ate half of the meat loaf that was there before Pedro stopped him What part of the meat loaf was still left? Use numbers, words, and/or pictures to solve the problem Show your work Answer: _ of the meat loaf was still left © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.49 A9.50 ã Bridges in Mathematics Grade Supplement â The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline For use anytime after Supplement Set A9, Activity NAME DATE Set A9 H Independent Worksheet INDEPENDENT WORKSHEET More Fraction Multiplication Fill in the chart to solve each of the problems below Multiplication Equation Word to Match Labeled Sketch ex 2×2= 3 Answer two-thirds of two-thirds a 2×6= b 1×4= c 3×4= Solve each problem 3×2= 4 1×3= 5×1= 6×3= 2×4= 6×1= 3ì1= 2ì2= â The Math Learning Center Bridges in Mathematics Grade Supplement • A9.51 A9.52 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline For use anytime after Supplement Set A9, Activity NAME DATE Set A9 H Independent Worksheet INDEPENDENT WORKSHEET Fraction Stories Jake is making cookies The recipe says he needs three-fourths of a cup of butter, but Jake wants to cut the recipe in half What is one-half of three-fourths of a cup of butter? Use numbers, words, and/or pictures to solve the problem Show your work Mrs Smith had 46 of a carton of eggs in her refrigerator She dropped the carton by accident and a fourth of the eggs in the carton broke How much of a carton of eggs did she have left after she cleaned up the mess? How many eggs was that? Use numbers, words, and/or pictures to solve the problem Show your work Write your own story problem to go with this expression Then solve it Use numbers, words, and/or pictures to solve the problem Show your work 1×2= CHALLENGE Rosa bought a bag of apples After she baked pies, she had 23 she gave her cousin have to start? © The Math Learning Center of a bag left Then of these, which was apples How many apples did Rosa Bridges in Mathematics Grade Supplement • A9.53 A9.54 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline For use anytime after Supplement Set A9, Activity NAME DATE Set A9 H Independent Worksheet INDEPENDENT WORKSHEET Using Strategies to Multiply Fractions with Mixed Numbers Use one of the strategies you know to multiply these problems: Example ì 45 ã Finding 45 of 1, times: 45 + 45 + 45 = 125 or 25 ã Findingthewholenumbertimesaunitfractionandscalingup:3ì 15 = 12 ì = ã Thinkingabout 45 as money or a decimal: 45 of a dollar equals $0.80 0.80 × = 2.40 a ; Nate is playing Target 1: Fractions He is trying to solve the following problem: 7× Solve Nate’s problem and show your work b Nate’s partner, Irie, solved × Show how you would solve it (Continued on next page.) © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.55 Set A9 Number & Operations: Multiplying Fractions Blackline NAME DATE Independent Worksheet Using Strategies to Multiply Fractions with Mixed Numbers (cont.) c Irie could have solved × instead of × Which problem is closer to 1? d Who will win this round? How you know? A9.56 ã Bridges in Mathematics Grade Supplement â The Math Learning Center Set A9 Number & Operations: Multiplying Fractions Blackline For use anytime after Supplement Set A9, Activity NAME DATE Set A9 H Independent Worksheet INDEPENDENT WORKSHEET Domino Multiplication Write the two fractions below the dominoes and then multiply them to find the product Show your work, and reduce the fraction if you can a b × = × = c d × = × = (Continued on next page.) © The Math Learning Center Bridges in Mathematics Grade Supplement • A9.57 Set A9 Number & Operations: Multiplying Fractions Blackline NAME DATE Independent Worksheet Domino Multiplication (cont.) Write a multiplication word problem that matches the following dominoes and solve it a b CHALLENGE Now invert one of the dominoes in each set to create a new improper fraction and then multiply the two fractions to find the product Remember to show your work! A9.58 • Bridges in Mathematics Grade Supplement © The Math Learning Center ... See Set A9 Independent Worksheets 1–3 on pages A9. 49? ?A9. 53 for more practice with multiplying fractions © The Math Learning Center Bridges in Mathematics Grade Supplement • A9. 27 Set A9 Number... Supplement Set A9 Independent Worksheet on pages A9. 55 and A9. 56 for more practice with multiplying fractions © The Math Learning Center Bridges in Mathematics Grade Supplement • A9. 33 Set A9 Number... Supplement Set A9 Independent Worksheet on pages A9. 57 and A9. 58 for more practice with multiplying fractions A9. 40 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A9 Number